Average Error: 18.3 → 1.2
Time: 31.5s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\left(\frac{v}{t1 + u} \cdot \frac{\sqrt[3]{-t1}}{\sqrt[3]{t1 + u}}\right) \cdot \frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\left(\frac{v}{t1 + u} \cdot \frac{\sqrt[3]{-t1}}{\sqrt[3]{t1 + u}}\right) \cdot \frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}
double f(double u, double v, double t1) {
        double r868100 = t1;
        double r868101 = -r868100;
        double r868102 = v;
        double r868103 = r868101 * r868102;
        double r868104 = u;
        double r868105 = r868100 + r868104;
        double r868106 = r868105 * r868105;
        double r868107 = r868103 / r868106;
        return r868107;
}
↓
double f(double u, double v, double t1) {
        double r868108 = v;
        double r868109 = t1;
        double r868110 = u;
        double r868111 = r868109 + r868110;
        double r868112 = r868108 / r868111;
        double r868113 = -r868109;
        double r868114 = cbrt(r868113);
        double r868115 = cbrt(r868111);
        double r868116 = r868114 / r868115;
        double r868117 = r868112 * r868116;
        double r868118 = r868114 * r868114;
        double r868119 = r868115 * r868115;
        double r868120 = r868118 / r868119;
        double r868121 = r868117 * r868120;
        return r868121;
}

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 18.3

    \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
  2. Using strategy rm

  3. Applied times-frac1.4

    \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt2.1

    \[\leadsto \frac{-t1}{\color{blue}{\left(\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}\right) \cdot \sqrt[3]{t1 + u}}} \cdot \frac{v}{t1 + u}\]

  6. Applied add-cube-cbrt1.7

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}\right) \cdot \sqrt[3]{-t1}}}{\left(\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}\right) \cdot \sqrt[3]{t1 + u}} \cdot \frac{v}{t1 + u}\]

  7. Applied times-frac1.7

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}} \cdot \frac{\sqrt[3]{-t1}}{\sqrt[3]{t1 + u}}\right)} \cdot \frac{v}{t1 + u}\]

  8. Applied associate-*l*1.2

    \[\leadsto \color{blue}{\frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}} \cdot \left(\frac{\sqrt[3]{-t1}}{\sqrt[3]{t1 + u}} \cdot \frac{v}{t1 + u}\right)}\]

  9. Final simplification1.2

    \[\leadsto \left(\frac{v}{t1 + u} \cdot \frac{\sqrt[3]{-t1}}{\sqrt[3]{t1 + u}}\right) \cdot \frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))